Modeling Energy Dissipation in a Tumbling Defunct ...Ryotaro Sakamoto and Daniel J. Scheeres...
Transcript of Modeling Energy Dissipation in a Tumbling Defunct ...Ryotaro Sakamoto and Daniel J. Scheeres...
Modeling Energy Dissipation in a Tumbling Defunct Satellite
using a Finite Element Method
Ryotaro Sakamoto and Daniel J. Scheeres
University of Colorado Boulder
429 UCB, Boulder, CO 80309
ABSTRACT
A detailed finite element model that captures energy dissipation effects on a tumbling satellite is presented. This
model is used to estimate the relaxation time of a satellite in a complex rotation mode, enabling comparison with
objects that have been observed to undergo rotational relaxation. This model can be used to develop better models
for energy dissipation in non-uniformly rotating debris and defunct satellites.
1. INTRODUCTION
The transition of defunct satellite rotation states from uniform rotation to tumbling has been predicted and observed
recently in the GOES-8 satellite [1]. These transitions are driven by solar radiation pressure torques, similar to the
so-called βYORP Effectβ which has been implicated in the evolution of asteroid rotation states [2]. The cycle
observed for GOES-8 apparently involves periods of spin-up about the minimum axis of inertia followed by a period
of energy dissipation, which brings the satellite back to rotation about its maximum moment of inertia. These
dynamics raise a number of interesting question about the evolution of spinning defunct satellites and debris in
general. In this research, we study one aspect of this cycle, focusing on the dissipation of energy. We develop
detailed models for energy dissipation in flexible satellites, using finite element calculations to model the flexible
dynamics of a satellite appendage subject to time-varying accelerations arising from a tumbling body. We use these
calculations to model the energy dissipation in a satellite appendage over every periodic cycle. The simulations
explore the amount of dissipation as a function of model parameters and as a function of the appendage orientation
relative to the satellites principal axes. Using the resulting dissipation and model parameters for the satellite, we feed
back the energy dissipation into the assumed torque-free rotation of the body to characterize the relaxation time
scale. The goal of this research is to develop models that will enable the estimation of energy dissipation on
observed satellites.
Our simulations model a simple solar array panel using a finite element method. Our models are two and three
dimensional. To evaluate the response of the model to a tumbling satellite, we model a time-periodic acceleration
profile distributed across the component, which has an additional offset from the ideal satellite center of mass. The
acceleration acting at a specific point on the appendage is found by combining a torque-free angular rate and
acceleration with the relative location of the component, and the finite element method computes the distributed
dynamical effect of these accelerations across the appendage. As the accelerations are periodic in the body-fixed
frame, the simulations are run over a long enough time for the behavior to settle into a steady oscillation. As the
local level of dissipation is increased, the amplitude of the oscillations become smaller. To determine the overall
effect on the body spin rate it is necessary to determine the total energy transferred into the appendage oscillation in
the absence of dissipation, comparing the peak energy to the dissipation over each cycle. To measure this
empirically we track the total amplitude of the steady-state oscillations as the dissipation in the appendage is
decreased. The following work demonstrate the results of the amount of the energy dissipation numerically. Each
result is considered with various spin rate and dumping coefficient. Numerical analyses are conducted as two and
three dimensional respectively. The method of the setting up these models are introduced briefly at the beginning of
the section. This allows the parameter Q to be computed, which can then be directly used to compute the energy
dissipation rate. This can be used to modify the spin state in a predictable way, allowing for the relaxation of the
excited spin state to be tracked over longer time spans, and evaluated at varying levels of excitation. These
calculations will be compared with observations of the tumbling satellite GOES-8 [1] to develop empirical estimates
of what appropriate level of dissipation should be used in our simulations for this particular asteroid.
Copyright Β© 2018 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS) β www.amostech.com
2. STRUCTUAL MODEL
In this section, we discuss the Finite Element Method and the analysis models applied in this research. FEM is used
as a tool to evaluate the material properties and their dissipation effectively [3]. As shown in Fig. 1 each element has
two nodes. L indicates the length of an element. Additionally, node coordinate matrices are also defined as in Eq. (1)
and (2). For two-dimensional analysis, the node coordinate has three components, X, Y, and the angle between
them. For three-dimensional analysis, the Z direction is added to X and Y and the resulting degrees of freedom are 6,
X, Y, Z, and the angles between them. The circled number in Fig. 1, indicates the node number.
Fig. 1. Elements and nodes number (Left: 2D, Right: 3D)
ππ¨ππ ππ¨π¨π«ππ’π§πππ (ππ) = (πΏ π π), ππ¨ππ ππππ«π’π± (ππ) = (π΅ππ πβ
π΅ππ πβ‘) = (
π π π
π π π) (1)
ππ¨ππ ππ¨π¨π«ππ’π§πππ (ππ) = (πΏ π π),
ππ¨ππ ππππ«π’π± (ππ) = (π΅ππ πβ
π΅ππ πβ‘) = (
π π π
π π π) (2)
Based on these fundamentals, we developed analysis models as shown in Fig. 2. The left model is for two-
dimensional analysis, the right one is for three-dimensional analysis with X Y Z axes. These setups are simulated for
a typical satellite model. cubes of each model indicate the main body component of satellites. The rightmost plates
model solar array panels. Each model has a total of seven or twenty-nine elements, respectively. In this research, the
material properties were set as aluminum, with parameters as shown in Tab. 1 [4]. As shown in this table, the Length
of one element is set at 1.0 [m]. Therefore, total size of two-dimensional model is with 4.0 m (X direction) Γ 2.0 m
(Y direction). The three-dimensional models has 1.0 m (X direction) Γ 4.0 m (Y direction) Γ 2.0 m (Z direction).
Parameters of the transverse module are only applied for the three-dimensional analysis.
Fig. 2. Analysis model (Left: 2D, Right: 3D)
Copyright Β© 2018 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS) β www.amostech.com
Tab. 1 Parameters of structural model
Parameter Value
Length of an element: L [m] 1.0
Thickness [m] 0.002
Width [m] 0.02
Density [kg/m3] 2857.4
Modulus of elasticity [GPa] 70.0
Transverse modulus of elasticity [GPa] 28.0
3. NUMERICAL ANALYSIS
3.1 Dynamical Equation
In this analysis, in order to investigate various system responses, damping was included in the simulations [5]. Here,
M is the mass matrix, C is the damping coefficient, K is the stiffness matrix, and X is state. This equation is a second
order differential equation and therefore was modified into a first order equation with X = x1, οΏ½ΜοΏ½= x2, as shown as Eq.
(4)
π΄οΏ½ΜοΏ½ + πͺοΏ½ΜοΏ½ + π²πΏ = π (3)
(π₯1Μ
π₯2Μ
) = (π 1
βπΎ
πβ
πΆ
π
) (π₯1
π₯2
) + (01
π
) πΉ (4)
For the two-dimensional FEM analysis, each X state and F matrix are defined as following. The X state has
difference information for x, y direction, and angle. The F matrix is described by the force along the x and y
direction.
π β π π‘ππ‘π = (
π·ππ πππππππππ‘(π₯)
π·ππ πππππππππ‘(π¦)
π΄ππππ
) , πΉ = (
π₯
π¦
ππππππ‘
) (5)
3.2 Rotational Acceleration Matrix
To model a satellite tumbling motion, we add rotational acceleration matrix as a time-varying acceleration. The
following equations present the two and three dimensional acceleration matrices. Here Οβ₯ represents a
perpendicular vector relative to the rotational axis.
For two-dimensional analysis,
π = (
π2π β πβ₯2 ππππ 2(π0π‘)
βπ Γ πβ₯2 sin (π0π‘)cos (π0π‘)
0
) (6)
Where, π0 =2π
3600, πβ₯ = πΌ Γ π0, π2 = π0
2 + πβ₯2
Copyright Β© 2018 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS) β www.amostech.com
For three-dimensional analysis, the r vector is given by node coordinate matrices in section 2.
π = (
π2 β πβ₯2 πππ 2(π0π‘) βπβ₯
2 π ππ(π0π‘)πππ (π0π‘) βπβ₯π0πππ (π0π‘)
βπβ₯2 sin (π0π‘)cos (π0π‘) π2 β πβ₯
2 π ππ2(π0π‘) βπβ₯π0π ππ(π0π‘)
βπβ₯π0πππ (π0π‘) βπβ₯π0π ππ(π0π‘) π2 β π02
) π (7)
Where, π = (
π₯
π¦
π§
)
Fig. 3 shows the exaggerated motion as the models are rotating. Left figure is shown when the two-dimensional
model is oscillating about the Y axis. Node 1 and 2 are considered as fixed. On the right is the three-dimensional
model oscillating about the X axis. As with the two-dimensional analysis, the origin of the model, node 1, is fixed.
Fig. 3. The images of the behaviors as model rotation (Left: 2D, Right: 3D)
4. SIMULATION RESULTS
4.1 Parameters and Overview
This section presents results of the calculations. Using these methods, as discussed above, we compute the energy
dissipation in several situations. We mainly change the rotational rate (Ο0) and damping coefficient (C) as free
parameters. Generally, attenuation of a system is caused by friction between the components of structure and
sloshing of liquid inside the system [6]. Also, structural attenuation is often discussed. However, in this study, we
did not focus on the factor of the attenuation. That is, we parametrize damping coefficient as a general attenuation.
For two-dimensional analysis, Eq (6) was taken into numerical simulation directly. Then, results are shown with the
modelsβ position difference, velocity, potential energy, and kinetic energy. These energies are calculated based on
following equations.
π·ππππππππ πππππ =π
ππΏππ²πΏ
π²ππππππ πππππ =π
ποΏ½ΜοΏ½ππ΄οΏ½ΜοΏ½ (8)
Copyright Β© 2018 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS) β www.amostech.com
4.2 Two-Dimensional analysis
Calculation results in two-dimension are shown in this section. Fig. 4 indicates the time history of the tip position
(Node 8 at Fig. 2 left) and velocities. Right side of Fig. 4 focuses on potential energy and kinetic energy. Each
colored line represents a damping coefficient ranging from 0.05 to 1. The rotational rate was set as 2Ο/300 [deg/sec].
We confirmed that the peak of each amplitude is decaying as damping gets larger. For two energies figure, the
difference of the peak of each amplitude, which means the amount of energy dissipation, depends on damping
coefficient. For instance, the purple line indicates the maximum energy with the damping coefficient is at a
minimum value. In contrast, the blue lineβs amplitude with damping coefficient equal to one, is lower than purpleβs.
Fig. 5 shows the tendency of the total energy dissipation with damping coefficient. Total energy is summed up
potential and kinetic energies. Plots were from the values each coefficient values. We confirmed two characteristics
about rate of energies dissipation. One is from the left side another is the right side in Fig. 5. The rate of energy
dissipation is coincidence even as the rotational rates are different. Second thing is that the rate of energy dissipation
is decreasing dramatically, as rotational speed gets faster.
To reach the concrete results, we plotted potential energy along horizontal axis versus positional difference of the tip
position at node 8 along on the vertical axis in Fig. 6. Rotational rate was set as 2Ο/600. From left up to right bottom,
the damping coefficient gets larger. Like hysteresis analysis, we calculated the area of each curve. Each figure in
Fig. 6 is one oscillation period respect to damping coefficient. The area help us understand the energy dissipation as
one cycle oscillation. Calculation results of area are shown in Fig. 7 rotational rates are 2Ο/600, 2Ο/1800, 2Ο/3600
respectively. From Fig. 6 and 7, we also confirmed the two tendency we mentioned above.
Fig. 4. Time history of the tip position (node 8) and energies
Copyright Β© 2018 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS) β www.amostech.com
Fig. 5. The rate of the energy dissipation with respect to the damping coefficient (2D)
Fig. 6. Area of Potential energy (Ο = 2Ο/600)
Copyright Β© 2018 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS) β www.amostech.com
Fig. 7. Area of potential energy respect to damping coefficient (2D)
4.3 Three-Dimensional Analysis
Similar to section 4.2, we also demonstrate the result of our three-dimensional calculations and confirm the tendency
between rotational rate and the amount of the energy dissipation. For three-dimensional analysis, we take X axis
rotation based on Eq. (7) to get reasonable results. We also could reach two tendencies in three-dimensional analysis
same as the as the two-dimensional ones. The first thing is that the amount of the energy dissipation gets larger as
the damping coefficient gets larger. The left side of Fig. 8 shows the position difference at the tip of the model,
which is node at 15 of the right side of Fig. 2. From up to bottom figure is showing X, Y, and Z axis respectively.
The right side of Fig. 8 is the time history of the potential and kinetic energies. Each colored line stands for the
damping coefficient. Rotational rate is set as 2Ο/300. Like two-dimensional analysis, we confirmed the energy
dissipation depends on the damping coefficient. The peaks of each oscillation get smaller as the damping coefficient
get larger. The amount of difference of the peak means the amount of energy dissipation we estimated.
The second thing is that the rate of the energy dissipation is faster as the rotational rate gets faster. Fig. 10Fig. 9
shows more detailed results, which have rotational rates from 2Ο/60 to 2Ο/1200. On the left side of Fig. 10, we
plotted the positioning difference along the X axis of the model and potential energy. Damping coefficient is also
compared in this figure. The right side of Fig. 10 shows the time history of the three-dimensional difference at node
15.
Fig. 8. Time history of the tip position (node 15) and energies
Copyright Β© 2018 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS) β www.amostech.com
Fig. 9. The rate of the energy dissipation with respect to damping coefficient (3D)
Fig. 10. Area of potential energy (Left) and position histry at node 16 (Right) (Ο0 = 2Ο/300)
Copyright Β© 2018 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS) β www.amostech.com
Fig. 11. Area of potential energy respect to damping coefficient (3D)
5. CONCLUSION
In order to calculate the energy dissipation rate in a tumbling satellite we quantitatively investigated the relationship
between energy dissipation, rotation rate, and damping coefficient, we established simple satellite models with
FEM. We computed the energy dissipation with two and three-dimension models. Several results were shown in
above section. The decay of the amplitude of the energies, considering the depend on the rotation rate and the
damping coefficient, were revealed. Additionally, numerical simulations for two and three dimensional
demonstrated the similar tendency for dissipating energy. We believe these results would contribute the way of
modifications the rotation rate or axis to stabilize the detumbling satellites or asteroids. However, these analyses
only offer simple axis rotation. To truly understand and predict the rotation state and the energy dissipation, more
complicated rotational acceleration modeling is needed. The next steps in this research will expand these models and
consider the effect of feeding back the computed level of dissipation into the evolving simulations.
6. REFERENCES
1. A. Albuja, D.J. Scheeres, R.L. Cognion, W. Ryan and E.V. Ryan. 2018. βThe YORP Effect on the GOES 8 and
GOES 10 Satellites: A Case Study,β Advances in Space Research 61: 122-144.
2. D.P. Rubincam. 2000. βRadiative spin-up and spin-down of small asteroids,β Icarus 148(1): 2-11.
3. Komatsu, K., Mechanical structure vibration engineering, Morikita Publishing Co., Ltd, Tokyo, Japan, 2012.
(in Japanese)
4. Sakamoto, R., Tanaka, H. and Ogi, Y., βImprovement of deployment repeatability by the vibration produced by
macro fiber composite actuatorβ, Bulletin of the JSME Mechanical Engineering Letters, Vol.2. No. 16-00042,
2016, pp.1-8.
5. Kojima, Y., Taniwaki, S., and Okami, Y., βDynamic simulation of stick-slip motion of a flexible solar arrayβ,
Control Engineering Practice, Vol. 16, 2008, pp. 724-735.
6. Tsuboi, M., βReport of the circumstances and the discontinuation of Astro -G projectβ, Japan VLBI
Consortium, Tokyo, Japan, 2011. (in Japanese).
Copyright Β© 2018 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS) β www.amostech.com